Integrate and Classify Normals · Getting started
Tools to integrate normal (Gaussian) distributions in any dimensions with any parameters within any domain, compute pdf/cdf/inverse cdf of any function of a normal vector, and compute quantities concerning classification performance among two or more multinormals, such as error matrix and discriminability 
.
.For function help, type:
doc integrate_normal
doc classify_normals
doc classify_normals_multi
doc norm_fun_cdf
doc norm_fun_pdf
doc norm_fun_inv
Abhranil Das, Center for Perceptual Systems, University of Texas at Austin
Bugs/questions/comments to abhranil.das@utexas.edu.
Integrate normal in 1d
Define normal parameters, and a quadratic function of it, 
, i.e. q(x)= -1*x^2 + 1*x + 1:
, i.e. q(x)= -1*x^2 + 1*x + 1:mu=-2; v=3;
quad.q2=-1;
quad.q1=1;
quad.q0=1;
Integrate the normal probability in the quadratic domain defined by 
:
p=integrate_normal(mu,v,quad)
p = 0.1941
Most plots can be zoomed and panned.
pdf/cdf/inverse cdf of 
p=norm_fun_cdf([0 1 2],mu,v,quad)
p = 1×3
0.8059 0.9175 1.0000
f=norm_fun_pdf([0 1 2],mu,v,quad)
f = 1×3
0.0865 0.1696 0
fplot(@(x) norm_fun_pdf(x,mu,v,quad),[-30 5]); ylabel 'pdf'
% 1st, 2nd and 3rd quartiles
x=norm_fun_inv([0.25 0.5 0.75],mu,v,quad)
x = 1×3
-12.2134 -5.0417 -0.7030
normal probability in the domain defined by a non-quadratic function 
fun=@(x) cos(x.^2);
[p,pc,bd_pts]=integrate_normal(mu,v,fun,'dom_type','fun','fun_span',5,'fun_resol',500)
p = 0.5803
pc = 0.4197
bd_pts = 1×100
-10.5606 -10.4108 -10.2588 -10.1045 -9.9479 -9.7887 -9.6269 -9.4623 -9.2948 -9.1243 -8.9505 -8.7732 -8.5923 -8.4075 -8.2185 -8.0251 -7.8269 -7.6236 -7.4147 -7.1997 -6.9782 -6.7493 -6.5124 -6.2666 -6.0107 -5.7434 -5.4631 -5.1675 -4.8541 -4.5189 -4.1568 -3.7599 -3.3160 -2.8025 -2.1708 -1.2533 1.2533 2.1708 2.8025 3.3160 3.7599 4.1568 4.5189 4.8541 5.1675 5.4631 5.7434 6.0107 6.2666 6.5124
pdf of 
x=linspace(-2,2,100);
f=norm_fun_pdf(x,mu,v,fun,'fun_span',5,'fun_resol',500,'dx',1e-2);
plot(x,f); ylabel 'pdf'
cdf of
F=norm_fun_cdf(x,mu,v,fun,'fun_span',5,'fun_resol',500);
plot(x,F); ylabel 'cdf'
Inverse cdf of
p=[0.5 0.75]; % median and 3rd quartile
x=norm_fun_inv(p,mu,v,fun)
x = 1×2
0.2803 0.8832
Classify in 1d
mu_1=0; v_1=1;
mu_2=2.5; v_2=1.5;
results=classify_normals([mu_1,v_1],[mu_2,v_2])
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [-11.2223 1.2223]
norm_errmat: [2×2 double]
norm_err: 0.1296
norm_dprime: 2.2565
norm_maha_dprime: 2.2361
with unequal priors and outcome values
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[3 0; 0 1])
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [-12.0987 2.0987]
norm_errmat: [2×2 double]
norm_err: 0.1240
norm_valmat: [2×2 double]
norm_val: 2.2509
Classify in 1d using samples
Priors are assumed prop. to sample sizes.
mu_1=0; v_1=1;
samp_1=normrnd(mu_1,sqrt(v_1),[700 1]);
mu_2=1.5; v_2=1.5;
samp_2=normrnd(mu_2,sqrt(v_2),[300 1]);
results=classify_normals(samp_1,samp_2,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -794
1 4 -798 initial simplex
2 5 -798 reflect
3 7 -799 reflect
4 9 -799 contract outside
5 10 -799 reflect
6 12 -800 contract inside
7 17 -800 shrink
8 19 -800 contract inside
9 24 -800 shrink
10 29 -800 shrink
11 31 -800 contract outside
12 36 -800 shrink
13 41 -800 shrink
14 46 -800 shrink
15 51 -800 shrink
16 56 -800 shrink
17 61 -800 shrink
18 66 -800 shrink
19 71 -800 shrink
20 76 -800 shrink
21 81 -800 shrink
22 86 -800 shrink
23 91 -800 shrink
24 96 -800 shrink
25 101 -800 shrink
26 106 -800 shrink
27 111 -800 shrink
28 116 -800 shrink
29 121 -800 shrink
30 126 -800 shrink
31 131 -800 shrink
32 136 -800 shrink
33 141 -800 shrink
34 146 -800 shrink
35 151 -800 shrink
36 156 -800 shrink
37 161 -800 shrink
38 166 -800 shrink
39 171 -800 shrink
40 176 -800 shrink
41 181 -800 shrink
42 186 -800 shrink
43 191 -800 shrink
44 196 -800 shrink
45 201 -800 shrink
46 206 -800 shrink
47 211 -800 shrink
48 216 -800 shrink
49 221 -800 shrink
50 226 -800 shrink
51 231 -800 shrink
52 236 -800 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-03
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [-6.3980 1.4393]
norm_errmat: [2×2 double]
norm_err: 0.2028
samp_errmat: [2×2 double]
samp_err: 0.2060
samp_dv: {[700×1 double] [300×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[700×1 double] [300×1 double]}
samp_opt_bd_pts: [-6.4294 1.5450]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.2000
Integrate normal in 2d
in a quadratic domain
defined by 
, i.e.
, i.e.q(x,y)= [x y]*[1 1; 1 1]*[x;y] + [-1 0]*[x;y] -1 > 0
mu=[-1; -1]; v=[1 0.5; 0.5 2];
quad.q2=[1 1; 1 1];
quad.q1=[-1;0];
quad.q0=-1;
% compare two integration algorithms
[p,pc,bd_pts]=integrate_normal(mu,v,quad) % ray method
p = 0.8214
pc = 0.1786
bd_pts = 2×504
103 ×
-0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0020 -0.0001 0.0020 -0.0001 0.0019 -0.0002 0.0019 -0.0002 0.0019 -0.0002 0.0018 -0.0002 0.0018 -0.0002 0.0018 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0016 -0.0002 0.0015 -0.0002 0.0015 -0.0002 0.0014 -0.0002 0.0013 -0.0003 0.0013
-0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0002 -0.0008 -0.0002 -0.0008 -0.0002 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0000 -0.0007 0.0000 -0.0007 0.0000 -0.0007 0.0001 -0.0006 0.0001 -0.0006 0.0002 -0.0006 0.0002
axis([-6 10 -10 2])
[p,pc]=integrate_normal(mu,v,quad,'method','gx2') % gx2 method
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.2e-03. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.2e-03. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
p = 0.8223
pc = 0.1777
axis([-6 10 -10 2])
PAPER In a non-quadratic domain
defined by 
:
:fun=@(x,y) x.*sin(y) - y.*cos(x) -1;
integrate_normal(mu,v,fun,'dom_type','fun','fun_span',3);
axis([-5 5 -5 5])
pdf of 
x=linspace(-10,10,100);
plot(x,norm_fun_pdf(x,mu,v,fun,'dx',1e-1))
PAPER probabilities of a function 
of a 2d normal
mu=[1;2]; v=.01*[1 1.6; 1.6 4];
fun=@(x,y) x.^y;
cdf
dx=.01; x=0:dx:3;
F=norm_fun_cdf(x,mu,v,fun,'fun_span',5,'fun_resol',500);
plot(x,F); xlabel x^y; ylabel 'cdf'
f=norm_fun_pdf(x,mu,v,fun,'fun_span',5,'fun_resol',500,'dx',1e-2);
plot(x,f); xlabel x^y; ylabel 'pdf'
Verify that pdf sums to 1:
sum(f)*dx
ans = 1.0000
Mean of 
m=sum(x.*f)*dx
m = 1.0266
sd of
m2=sum(x.^2.*f)*dx;
sd=sqrt(m2-m^2)
sd = 0.2115
median of
norm_fun_inv(.5,mu,v,fun)
ans = 1.0000
Integrate joint pdf of functions of a normal
% (x,y) is a normal vector with these parameters:
mu=[6;-19];
v=[1 -.7; -.7 2];
% functions of the normal vector
fun1=@(x,y) cos(4*x);
fun2=@(x,y) cos(4*y);
% p(cos 4x > 0)
integrate_normal(mu,v,fun1,'dom_type','fun','fun_span',3);
axis image; axis([2 10 -27 -11])
% p(cos 4y > 0)
integrate_normal(mu,v,fun2,'dom_type','fun','fun_span',3);
axis image; axis([2 10 -27 -11])
Probability that 
and 
are both > 0, i.e. integrate the normal in the domain 
:
and are both > 0, i.e. integrate the normal in the domain :fun3=@(x,y) min(fun1(x,y),fun2(x,y));
p=integrate_normal(mu,v,fun3,'dom_type','fun','fun_span',3)
p = 0.2495
axis image; axis([2 10 -27 -11])
Probability that 
, i.e. integrate the normal in the domain 
, i.e. integrate the normal in the domain fun4=@(x,y) fun1(x,y)+fun2(x,y)-0.5;
p=integrate_normal(mu,v,fun4,'dom_type','fun','fun_span',3)
p = 0.3072
axis image; axis([2 10 -27 -11]);
Percentage points (inverse cdf):
p=[0.5 0.75];
x=norm_fun_inv(p,mu,v,fun4)
x = 1×2
-0.4999 0.2210
Classify two 2d normals, one inside the other
mu_1=[4; 5]; v_1=[2 1; 1 1];
mu_2=mu_1; v_2=3*[2 -1; -1 1];
% compare two integration algorithms
results_ray=classify_normals([mu_1,v_1],[mu_2,v_2]) % ray method
results_ray = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [2×414 double]
norm_errmat: [2×2 double]
norm_err: 0.1979
norm_dprime: 1.6982
norm_maha_dprime: 0
axis([-2 8 0 10])
results_gx2=classify_normals([mu_1,v_1],[mu_2,v_2],'method','gx2') % gx2 method
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.7e-06. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.7e-06. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
results_gx2 = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1979
norm_dprime: 1.6982
norm_maha_dprime: 0
axis([-2 8 0 10])
PAPER Classify in 2d with custom boundaries and from samples
mu_1=[2;4]; v_1=[1 1.5; 1.5 3];
mu_2=[5;0]; v_2=[3 0; 0 1];
% ray method
results_ray=classify_normals([mu_1,v_1],[mu_2,v_2]);
axis image; axis([-10 10 -10 10])
results_ray.norm_err
ans = 0.0079
% compare with generalized chi square method
results_gx2=classify_normals([mu_1,v_1],[mu_2,v_2],'method','gx2');
axis image; axis([-10 10 -10 10])
results_gx2.norm_err
ans = 0.0079
% now supply a custom linear boundary
linear_bd.q2=zeros(2);
linear_bd.q1=[-.7;1];
linear_bd.q0=0;
results_linear=classify_normals([mu_1,v_1],[mu_2,v_2],'dom',linear_bd)
results_linear = struct with fields:
norm_bd_pts: [2×300 double]
norm_errmat: [2×2 double]
norm_err: 0.0133
axis image; axis([-10 10 -10 10])
% now classify using samples
n_samp=1e3;
samp_1=mvnrnd(mu_1,v_1,n_samp);
samp_2=mvnrnd(mu_2,v_2,n_samp);
results_samp=classify_normals(samp_1,samp_2,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -1985
1 7 -1985 initial simplex
2 9 -1985 contract inside
3 11 -1985 contract outside
4 19 -1985 shrink
5 27 -1985 shrink
6 35 -1985 shrink
7 43 -1985 shrink
8 51 -1985 shrink
9 59 -1985 shrink
10 67 -1985 shrink
11 75 -1985 shrink
12 83 -1985 shrink
13 91 -1985 shrink
14 99 -1985 shrink
15 107 -1985 shrink
16 115 -1985 shrink
17 123 -1985 shrink
18 131 -1985 shrink
19 139 -1985 shrink
20 147 -1985 shrink
21 155 -1985 shrink
22 163 -1985 shrink
23 171 -1985 shrink
24 179 -1985 shrink
25 187 -1985 shrink
26 195 -1985 shrink
27 203 -1985 shrink
28 211 -1985 shrink
29 219 -1985 shrink
30 227 -1985 shrink
31 235 -1985 shrink
32 243 -1985 shrink
33 251 -1985 shrink
34 259 -1985 shrink
35 267 -1985 shrink
36 275 -1985 shrink
37 283 -1985 shrink
38 291 -1985 shrink
39 299 -1985 shrink
40 307 -1985 shrink
41 315 -1985 shrink
42 323 -1985 shrink
43 331 -1985 shrink
44 339 -1985 shrink
45 347 -1985 shrink
46 355 -1985 shrink
47 363 -1985 shrink
48 371 -1985 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results_samp = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [2×508 double]
norm_errmat: [2×2 double]
norm_err: 0.0077
norm_dprime: 4.8496
norm_maha_dprime: 4.4594
samp_errmat: [2×2 double]
samp_err: 0.0075
samp_dprime: 4.8648
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd_pts: [2×31650 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.0075
samp_opt_dprime: 4.8648
axis image; axis([-10 10 -10 10])
% modify the sample-optimized boundary and classify with it
custom_bd=results_samp.samp_opt_bd;
custom_bd.q2=custom_bd.q2+.1;
custom_bd.q1=custom_bd.q1-2.5;
custom_bd.q0=custom_bd.q0+5;
results_samp_custom=classify_normals(samp_1,samp_2,'input_type','samp','dom',custom_bd)
Iteration Func-count min f(x) Procedure
0 1 -1946
1 7 -1960 initial simplex
2 8 -1960 reflect
3 10 -1975 expand
4 11 -1975 reflect
5 12 -1975 reflect
6 14 -1980 expand
7 15 -1980 reflect
8 17 -1981 reflect
9 18 -1981 reflect
10 19 -1981 reflect
11 20 -1981 reflect
12 22 -1981 contract outside
13 30 -1981 shrink
14 38 -1981 shrink
15 40 -1981 contract outside
16 48 -1981 shrink
17 50 -1982 reflect
18 58 -1982 shrink
19 66 -1982 shrink
20 74 -1982 shrink
21 75 -1982 reflect
22 77 -1982 contract inside
23 78 -1982 reflect
24 80 -1982 contract outside
25 88 -1982 shrink
26 96 -1982 shrink
27 104 -1982 shrink
28 112 -1982 shrink
29 120 -1982 shrink
30 128 -1982 shrink
31 136 -1982 shrink
32 144 -1982 shrink
33 152 -1982 shrink
34 160 -1982 shrink
35 168 -1982 shrink
36 176 -1982 shrink
37 184 -1982 shrink
38 192 -1982 shrink
39 200 -1982 shrink
40 208 -1982 shrink
41 216 -1982 shrink
42 224 -1982 shrink
43 232 -1982 shrink
44 240 -1982 shrink
45 248 -1982 shrink
46 256 -1982 shrink
47 264 -1982 shrink
48 272 -1982 shrink
49 280 -1982 shrink
50 288 -1982 shrink
51 296 -1982 shrink
52 304 -1982 shrink
53 312 -1982 shrink
54 320 -1982 shrink
55 328 -1982 shrink
56 336 -1982 shrink
57 344 -1982 shrink
58 352 -1982 shrink
59 360 -1982 shrink
60 368 -1982 shrink
61 376 -1982 shrink
62 384 -1982 shrink
63 392 -1982 shrink
64 400 -1982 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results_samp_custom = struct with fields:
norm_bd_pts: [2×414 double]
norm_errmat: [2×2 double]
norm_err: 0.0225
samp_errmat: [2×2 double]
samp_err: 0.0270
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd_pts: [2×26576 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.0090
axis image; axis([-10 10 -10 10])
PAPER Classify in 2d with non-normal samples
n_samp=1e3;
samp_1=exp(mvnrnd([0 0],eye(2),n_samp));
samp_2=-exp(mvnrnd([1 1],eye(2),n_samp))+[2 2];
results=classify_normals(samp_1,samp_2,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -1477
1 7 -1481 initial simplex
2 9 -1490 expand
3 10 -1490 reflect
4 11 -1490 reflect
5 13 -1495 expand
6 14 -1495 reflect
7 16 -1513 expand
8 17 -1513 reflect
9 19 -1529 expand
10 20 -1529 reflect
11 22 -1556 expand
12 23 -1556 reflect
13 25 -1607 expand
14 26 -1607 reflect
15 28 -1666 expand
16 29 -1666 reflect
17 31 -1727 expand
18 32 -1727 reflect
19 34 -1730 reflect
20 35 -1730 reflect
21 36 -1730 reflect
22 38 -1730 contract inside
23 40 -1730 contract inside
24 42 -1730 contract outside
25 44 -1730 contract inside
26 46 -1730 contract inside
27 48 -1730 contract inside
28 50 -1731 contract inside
29 52 -1731 contract outside
30 54 -1736 contract inside
31 56 -1736 contract outside
32 58 -1736 contract inside
33 60 -1736 contract inside
34 62 -1737 contract inside
35 64 -1741 reflect
36 66 -1741 contract inside
37 67 -1741 reflect
38 69 -1741 contract inside
39 70 -1741 reflect
40 71 -1741 reflect
41 73 -1741 contract outside
42 75 -1741 contract inside
43 76 -1741 reflect
44 77 -1741 reflect
45 79 -1742 reflect
46 80 -1742 reflect
47 82 -1743 reflect
48 83 -1743 reflect
49 85 -1743 contract inside
50 86 -1743 reflect
51 88 -1745 reflect
52 90 -1745 contract inside
53 92 -1748 expand
54 93 -1748 reflect
55 95 -1748 contract inside
56 96 -1748 reflect
57 98 -1748 contract outside
58 100 -1753 expand
59 101 -1753 reflect
60 102 -1753 reflect
61 103 -1753 reflect
62 105 -1755 expand
63 107 -1758 reflect
64 108 -1758 reflect
65 109 -1758 reflect
66 111 -1759 reflect
67 112 -1759 reflect
68 114 -1761 reflect
69 115 -1761 reflect
70 117 -1761 contract inside
71 125 -1761 shrink
72 126 -1761 reflect
73 134 -1761 shrink
74 135 -1761 reflect
75 143 -1761 shrink
76 151 -1762 shrink
77 153 -1762 contract outside
78 161 -1762 shrink
79 162 -1762 reflect
80 163 -1762 reflect
81 164 -1762 reflect
82 165 -1762 reflect
83 173 -1762 shrink
84 175 -1762 contract outside
85 177 -1762 contract inside
86 185 -1762 shrink
87 193 -1762 shrink
88 201 -1762 shrink
89 209 -1762 shrink
90 217 -1762 shrink
91 225 -1762 shrink
92 233 -1762 shrink
93 241 -1762 shrink
94 249 -1762 shrink
95 257 -1762 shrink
96 265 -1762 shrink
97 273 -1762 shrink
98 281 -1762 shrink
99 289 -1762 shrink
100 297 -1762 shrink
101 305 -1762 shrink
102 313 -1762 shrink
103 321 -1762 shrink
104 329 -1762 shrink
105 337 -1762 shrink
106 345 -1762 shrink
107 353 -1762 shrink
108 361 -1762 shrink
109 369 -1762 shrink
110 377 -1762 shrink
111 385 -1762 shrink
112 393 -1762 shrink
113 401 -1762 shrink
114 409 -1762 shrink
115 417 -1762 shrink
116 425 -1762 shrink
117 433 -1762 shrink
118 441 -1762 shrink
119 449 -1762 shrink
120 457 -1762 shrink
121 465 -1762 shrink
122 473 -1762 shrink
123 481 -1762 shrink
124 489 -1762 shrink
125 497 -1762 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [2×482 double]
norm_errmat: [2×2 double]
norm_err: 0.1253
norm_dprime: 2.2980
norm_maha_dprime: 1.3247
samp_errmat: [2×2 double]
samp_err: 0.2615
samp_dprime: 1.2775
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd_pts: [2×29246 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.1190
samp_opt_dprime: 2.3600
axis image; axis([-20 15 -20 15])
Inversion/union/intersection of integration/classification domains
mu=[0;0]; v=[.5 0; 0 1];
circle_left=@(x,y) -(x+1).^2-y.^2+5;
circle_right=@(x,y) -(x-1).^2-y.^2+5;
circle_union=@(x,y) max(circle_left(x,y), circle_right(x,y));
p=integrate_normal(mu,v,circle_union,'dom_type','fun','fun_span',5)
p = 0.9680
axis image; axis([-4 4 -4 4])
circle_intersection=@(x,y) min(circle_left(x,y), circle_right(x,y));
p=integrate_normal(mu,v,circle_intersection,'dom_type','fun','fun_span',5)
p = 0.7949
axis image; axis([-4 4 -4 4])
crescent=@(x,y) max(circle_left(x,y), -circle_right(x,y));
p=integrate_normal(mu,v,crescent,'dom_type','fun','fun_span',5)
p = 0.9136
axis image; axis([-4 4 -4 4])
% classify normals using this crescent domain
mu_2=[2.2;0]; v_2=[.5 0; 0 .25];
results=classify_normals([mu,v],[mu_2,v_2],'dom',crescent,'dom_type','fun')
results = struct with fields:
norm_bd_pts: [2×586 double]
norm_errmat: [2×2 double]
norm_err: 0.1235
axis image; axis([-2 4 -3 3])
Classify in 3d
mu_1=[0;0;0]; v_1=eye(3);
mu_2=[2;1;1]; v_2=2*eye(3);
results=classify_normals([mu_1,v_1],[mu_2,v_2])
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [3×11936 double]
norm_errmat: [2×2 double]
norm_err: 0.1390
norm_dprime: 2.1694
norm_maha_dprime: 2
Estimate tiny errors (large 
) with high accuracy
format long
dprime_true=75
dprime_true =
75
mu_1=[0;0;0]; v=eye(3);
mu_2=dprime_true*[1;0;0];
results=classify_normals([mu_1,v],[mu_2,v],'AbsTol',0,'RelTol',1e-2)
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [3×14112 double]
norm_errmat: [2×2 double]
norm_err: 4.608155172364448e-308
norm_dprime: 74.999967581801229
norm_maha_dprime: 75
axis image; axis([-5 80 -2 2 -2 2]); view(-53,25)
dprime_computed=results.norm_dprime
dprime_computed =
74.999967581801229
format
PAPER accuracy vs separation between the normals
mu_1=[0;0;0];
% both normals have the same covariance, so we can check against
% the true d' (Mahalanobis distance).
v=[1 .5 .7;
.5 2 1 ;
.7 1 3];
steps=linspace(1,100,10);
d_true=nan(size(steps));
d_gx2=nan(size(steps));
d_ray=nan(size(steps));
parfor i=1:length(steps)
i
mu_2=steps(i)*[1;1;1];
results_gx2=classify_normals([mu_1,v],[mu_2,v],'method','gx2','AbsTol',0,'RelTol',0,'plotmode',false);
d_true(i)=results_gx2.norm_maha_dprime;
d_gx2(i)=results_gx2.norm_dprime;
results_ray=classify_normals([mu_1,v],[mu_2,v],'method','ray','AbsTol',0,'RelTol',0,'plotmode',false);
d_ray(i)=results_ray.norm_dprime;
end
Starting parallel pool (parpool) using the 'local' profile ...
Connected to the parallel pool (number of workers: 4).
ans =
1
ans =
2
ans =
3
ans =
4
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 45)
In integrate_normal (line 85)
In classify_normals (line 191)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 44)
In integrate_normal (line 85)
In classify_normals (line 193)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
ans =
10
ans =
9
ans =
8
ans =
7
ans =
6
ans =
5
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 45)
In integrate_normal (line 85)
In classify_normals (line 191)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 44)
In integrate_normal (line 85)
In classify_normals (line 193)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
rel_err_gx2=abs(d_gx2-d_true)./d_true;
rel_err_ray=abs(d_ray-d_true)./d_true;
figure; hold on
plot(d_true,rel_err_gx2,'-o')
plot(d_true,rel_err_ray,'-o')
xline(-2*norminv(realmin)) % largest computable d', corr. to the smallest possible error representable in double-precision
yline(eps) % machine epsilon for double precision
xlim([1 80]); ylim([0 2.2e-15])
xlabel 'true d'''
ylabel('$\epsilon$','interpreter','latex')
legend({'$\tilde{\chi}^2$','ray'},'interpreter','latex')
PAPER Integrate normal in a torus defined by function 
mu=[0;0;0];
v=[1 0 0;
0 8 4;
0 4 8];
% plot the error ellipsoid of the normal
figure; plot_normal(mu,v);
fun_torus=@(x1,x2,x3) 1.5-(5-(x1.^2+x2.^2).^0.5).^2-x3.^2;
integrate_normal(mu,v,fun_torus,'dom_type','fun','fun_span',3,'fun_resol',10,'RelTol',1e-1);
axis image; axis([-7 7 -7 7 -7 7])
Plot the pdf and cdf of
x=[linspace(-80,-20,30) linspace(-20,10,20)];
f=norm_fun_pdf(x,mu,v,fun_torus,'fun_span',3,'fun_resol',10,'RelTol',1e-1,'dx',8);
plot(x,f)
p=norm_fun_cdf(x,mu,v,fun_torus,'fun_span',3,'fun_resol',10,'RelTol',1e-1);
plot(x,p)
PAPER Integrate normal in 4d
outside the sphere of radius 5, i.e. 
, i.e. x'*eye(4)*x + zeros(4,1)'*x -25 >0
, i.e. x'*eye(4)*x + zeros(4,1)'*x -25 >0mu=[1;1;1;1]; v=diag([1 2 3 4]);
quad.q2=eye(4);
quad.q1=zeros(4,1);
quad.q0=-25;
p=integrate_normal(mu,v,quad)
p = 0.1276
axis([-30 40 0 .06]);
PAPER Classify in 4d, with priors and outcome values
mu_1=[0;0;0;0]; v_1=eye(4);
mu_2=[1;1;1;1]; v_2=eye(4);
% here the Bayes decision variable is normally distributed
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[2 0; 0 1])
results = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1496
norm_valmat: [2×2 double]
norm_val: 1.5236
mu_1=[0;0;0;0];
v_1=[1 0 0 0;
0 2 -1 0;
0 -1 3 2;
0 0 2 4];
mu_2=1.5*[1;1;1;1];
v_2=[2 0 0 0;
0 2 0 0;
0 0 2 0;
0 0 0 1];
% correct classification of class 1 is valued 4x than class 2
% here the Bayes decision variable is distributed as a generalized chi-square
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[4 0; 0 1])
results = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1302
norm_valmat: [2×2 double]
norm_val: 2.9431
% use ray method (with Monte Carlo integration)
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[4 0; 0 1],'method','ray','mc_samples',1e3)
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [4×2432 double]
norm_errmat: [2×2 double]
norm_err: 0.1284
norm_valmat: [2×2 double]
norm_val: 2.9470
% now classify using samples
n_samp=1e3;
results=classify_normals(mvnrnd(mu_1',v_1,n_samp),mvnrnd(mu_2',v_2,n_samp),'input_type','samp','prior_1',.7,'vals',[4 0; 0 1])
Iteration Func-count min f(x) Procedure
0 1 -4547
1 16 -4560 initial simplex
2 18 -4577 reflect
3 19 -4577 reflect
4 20 -4577 reflect
5 21 -4577 reflect
6 22 -4577 reflect
7 23 -4577 reflect
8 24 -4577 reflect
9 25 -4577 reflect
10 26 -4577 reflect
11 27 -4577 reflect
12 28 -4577 reflect
13 29 -4577 reflect
14 30 -4577 reflect
15 31 -4577 reflect
16 33 -4581 expand
17 34 -4581 reflect
18 36 -4583 reflect
19 37 -4583 reflect
20 38 -4583 reflect
21 39 -4583 reflect
22 40 -4583 reflect
23 41 -4583 reflect
24 42 -4583 reflect
25 43 -4583 reflect
26 44 -4583 reflect
27 45 -4583 reflect
28 47 -4585 reflect
29 48 -4585 reflect
30 49 -4585 reflect
31 50 -4585 reflect
32 51 -4585 reflect
33 53 -4585 contract inside
34 54 -4585 reflect
35 55 -4585 reflect
36 72 -4585 shrink
37 73 -4585 reflect
38 74 -4585 reflect
39 75 -4585 reflect
40 77 -4585 contract inside
41 78 -4585 reflect
42 79 -4585 reflect
43 80 -4585 reflect
44 82 -4585 contract inside
45 84 -4585 contract inside
46 86 -4585 contract inside
47 88 -4585 contract inside
48 90 -4587 reflect
49 92 -4587 contract inside
50 94 -4587 contract inside
51 96 -4587 contract inside
52 97 -4587 reflect
53 99 -4587 contract inside
54 100 -4587 reflect
55 101 -4587 reflect
56 103 -4587 contract inside
57 105 -4587 contract inside
58 106 -4587 reflect
59 107 -4587 reflect
60 109 -4587 contract inside
61 111 -4587 contract inside
62 113 -4588 reflect
63 115 -4588 contract inside
64 117 -4588 contract inside
65 118 -4588 reflect
66 119 -4588 reflect
67 121 -4588 contract inside
68 122 -4588 reflect
69 124 -4588 contract inside
70 125 -4588 reflect
71 127 -4588 contract inside
72 128 -4588 reflect
73 129 -4588 reflect
74 130 -4588 reflect
75 131 -4588 reflect
76 148 -4588 shrink
77 165 -4588 shrink
78 166 -4588 reflect
79 167 -4588 reflect
80 168 -4588 reflect
81 169 -4588 reflect
82 170 -4588 reflect
83 187 -4588 shrink
84 188 -4588 reflect
85 190 -4588 contract inside
86 207 -4588 shrink
87 224 -4588 shrink
88 241 -4588 shrink
89 258 -4588 shrink
90 275 -4588 shrink
91 292 -4588 shrink
92 309 -4588 shrink
93 326 -4588 shrink
94 343 -4588 shrink
95 360 -4588 shrink
96 377 -4588 shrink
97 394 -4588 shrink
98 411 -4588 shrink
99 428 -4588 shrink
100 445 -4588 shrink
101 462 -4588 shrink
102 479 -4588 shrink
103 496 -4588 shrink
104 513 -4588 shrink
105 530 -4588 shrink
106 547 -4588 shrink
107 564 -4588 shrink
108 581 -4588 shrink
109 598 -4588 shrink
110 615 -4588 shrink
111 632 -4588 shrink
112 649 -4588 shrink
113 666 -4588 shrink
114 683 -4588 shrink
115 700 -4588 shrink
116 717 -4588 shrink
117 734 -4588 shrink
118 751 -4588 shrink
119 768 -4588 shrink
120 785 -4588 shrink
121 802 -4588 shrink
122 819 -4588 shrink
123 836 -4588 shrink
124 853 -4588 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1311
norm_valmat: [2×2 double]
norm_val: 2.9424
samp_errmat: [2×2 double]
samp_err: 0.2145
samp_valmat: [2×2 double]
samp_val: 4547
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.1895
samp_opt_valmat: [2×2 double]
samp_opt_val: 4588
PAPER Integrate in a 4d polyhedron, using Monte Carlo
mu=zeros(4,1);
v=eye(4);
fun=@(x1,x2,x3,x4) abs(x1)+abs(x2)+abs(x3)+abs(x4);
% find p(f(x)<1)
mc_samples=round(10.^linspace(1,4,20)); % # of Monte Carlo samples
n_repeat=5;
plist=nan(length(mc_samples),n_repeat);
for k=1:n_repeat
for i=1:length(mc_samples)
[~,p]=integrate_normal(mu,v,fun,'dom_type','fun','fun_level',1,'fun_span',3,'fun_resol',10,'mc_samples',mc_samples(i),'plotmode',0);
plist(i,k)=p;
end
end
plot(mc_samples,plist,'-k','marker','.','markersize',10)
xlabel 'Monte Carlo sample size'; ylabel 'p'
set(gca,'xscale','log')
Compute cdf and pdf of the polyhedral function
x=linspace(-1,8,50);
p=norm_fun_cdf(x,mu,v,fun);
plot(x,p,'-o')
f=norm_fun_pdf(x,mu,v,fun,'dx',0.5,'mc_samples',5e3);
plot(x,f)
Classifying 3 1d normals
normals=struct;
normals(1).mu=-1; normals(1).v=1;
normals(2).mu=1; normals(2).v=2;
normals(3).mu=4; normals(3).v=.5;
% priors and outcome values
priors=[.4 .5 .1];
vals=[4 0 0; 0 1 -2; 0 0 1];
results=classify_normals_multi(normals,'priors',priors,'vals',vals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
results = struct with fields:
norm_bd_pts: {3×1 cell}
norm_errmat: [3×3 double]
norm_err: 0.2834
norm_valmat: [3×3 double]
norm_val: 1.8466
Classifying 4 2d normals
% define struct of all normals
normals=struct;
normals(1).mu=[1;0]; normals(1).v=2*eye(2);
normals(2).mu=[0;1]; normals(2).v=eye(2);
normals(3).mu=[-1;0]; normals(3).v=eye(2);
normals(4).mu=[0;-1]; normals(4).v=eye(2);
% plot the multi-class boundary of normal 3
plot_boundary(@(n,mu,v) opt_class_multi(n,normals,3,'mu',mu,'v',v),2,'mu',normals(3).mu,'dom_type','ray_scan')
title 'boundary of normal 3'
% classify
results=classify_normals_multi(normals); axis image
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
% now classify using samples from these normals, with outcome values
samples=struct;
for i=1:4
samples(i).sample=mvnrnd(normals(i).mu,normals(i).v,1e4);
end
vals=diag([1 2 3 4]);
results_samp=classify_normals_multi(samples,'input_type','samp','vals',vals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results_samp = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.4610
norm_valmat: [4×4 double]
norm_val: 1.5608
samp_errmat: [4×4 double]
samp_err: 0.4626
samp_valmat: [4×4 double]
samp_val: 62324
axis image
% now classify using samples from t distributions similar to these normals
samples=struct;
for i=1:4
samples(i).sample=mvtrnd(normals(i).v,3,1e4)+normals(i).mu';
end
results_samp=classify_normals_multi(samples,'input_type','samp')
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results_samp = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.5709
samp_errmat: [4×4 double]
samp_err: 0.4575
axis image; axis([-6 6 -6 6])
PAPER Classifying 7 2d normals
% define struct of all normals
normals=struct;
normals(1).mu=[2;0]; normals(1).v=[2 1; 1 2];
normals(2).mu=[0;1]; normals(2).v=[.5 0; 0 1];
normals(3).mu=[-1;0]; normals(3).v=[1 .3; .3 1];
normals(4).mu=[0;-1]; normals(4).v=[.5 -.5; -.5 1];
normals(5).mu=[-2;2]; normals(5).v=.5*[1 1; 1 5];
normals(6).mu=[2;-3]; normals(6).v=.3*eye(2);
normals(7).mu=[-2;-2.5]; normals(7).v=[1 0; 0 .1];
% classify
results=classify_normals_multi(normals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 5
Integrating normal 1 in domain 6
Integrating normal 1 in domain 7
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 5
Integrating normal 2 in domain 6
Integrating normal 2 in domain 7
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 5
Integrating normal 3 in domain 6
Integrating normal 3 in domain 7
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 5
Integrating normal 4 in domain 6
Integrating normal 4 in domain 7
Integrating normal 5 in domain 1
Integrating normal 5 in domain 2
Integrating normal 5 in domain 3
Integrating normal 5 in domain 4
Integrating normal 5 in domain 5
Integrating normal 5 in domain 6
Integrating normal 5 in domain 7
Integrating normal 6 in domain 1
Integrating normal 6 in domain 2
Integrating normal 6 in domain 3
Integrating normal 6 in domain 4
Integrating normal 6 in domain 5
Integrating normal 6 in domain 6
Integrating normal 6 in domain 7
Integrating normal 7 in domain 1
Integrating normal 7 in domain 2
Integrating normal 7 in domain 3
Integrating normal 7 in domain 4
Integrating normal 7 in domain 5
Integrating normal 7 in domain 6
Integrating normal 7 in domain 7
results = struct with fields:
norm_bd_pts: {7×1 cell}
norm_errmat: [7×7 double]
norm_err: 0.2319
axis image; axis([-10 10 -10 10]);
Classifying 4 3d normals
normals=struct;
normals(1).mu=[1;0;0]; normals(1).v=2*eye(3);
normals(2).mu=[0;1;0]; normals(2).v=eye(3);
normals(3).mu=[-1;0;0]; normals(3).v=eye(3);
normals(4).mu=[0;-1;0]; normals(4).v=eye(3);
results=classify_normals_multi(normals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.4254
PAPER Classifying 4 4d t distribution samples
params=struct;
params(1).mu=[0;-4;-2;-2]; params(1).v=diag([1 2 3 4]);
params(2).mu=[1;1;0;0]; params(2).v=eye(4)/4;
params(3).mu=[3;4;5;7]; params(3).v=eye(4);
params(4).mu=[5;5;7;4]; params(4).v=eye(4);
priors=[.3 .5 .15 .05];
n_samp=1e3;
samples=struct;
for i=1:4
samples(i).sample=mvtrnd(params(i).v,3,n_samp)+params(i).mu';
end
results_samp=classify_normals_multi(samples,'input_type','samp','priors',priors,'mc_samples',1e3,'plotmode',[1;1;1;1])
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results_samp = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.0402
samp_errmat: [4×4 double]
samp_err: 0.0580
PAPER testing the normal approximation for 2 2d classes
N_samp=1e4; n_samp=1e2;
mu_1=[5 5]; v_1=[1 .5; .5 1]; samp_1=mvnrnd(mu_1,v_1,N_samp).^3; % not normal
mu_2=[-200 -200]; v_2=[1 1; 1 3]*2e3; samp_2=mvnrnd(mu_2,v_2,N_samp);
results=classify_normals(samp_1,samp_2,'input_type','samp','samp_opt',false);
axis([-300 600 -400 600]); hold on
bd=results.norm_bd;
q0s=linspace(-100,35,100)';
true_p11=nan(size(q0s)); true_p2_2=nan(size(q0s)); % p(1|1)
norm_p11=nan(size(q0s)); norm_p2_2=nan(size(q0s));
for i=1:length(q0s)
bd_shift=bd; bd_shift.q0=q0s(i);
if ~rem(i,10)
plot_boundary(bd_shift,2,'plot_type','line','line_color',.5*[1 1 1]);
end
results_shift=classify_normals(samp_1,samp_2,'input_type','samp','dom',bd_shift,'samp_opt',false,'plotmode',false);
% true outcomes
true_p11(i)=results_shift.samp_errmat(1,1)/N_samp;
true_p2_2(i)=results_shift.samp_errmat(2,2)/N_samp;
% normal outcomes
norm_p11(i)=results_shift.norm_errmat(1,1)/.5;
norm_p2_2(i)=results_shift.norm_errmat(2,2)/.5;
end
true_p1_1_sd=sqrt(true_p11.*(1-true_p11)/n_samp);
true_p2_2_sd=sqrt(true_p2_2.*(1-true_p2_2)/n_samp);
norm_p1_1_sd=sqrt(norm_p11.*(1-norm_p11)/n_samp);
norm_p2_2_sd=sqrt(norm_p2_2.*(1-norm_p2_2)/n_samp);
figure; hold on
colors=colororder;
xline(bd.q0,'k','optimal boundary','LabelVerticalAlignment','bottom') % optimal criterion
% p(1|1)
x=q0s; y=true_p11; dy=true_p1_1_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],colors(1,:),'facealpha',.3,'linestyle','none');
y=norm_p11; dy=norm_p1_1_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',colors(1,:));
% p(2|2)
y=true_p2_2; dy=true_p2_2_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],colors(2,:),'facealpha',.3,'linestyle','none');
y=norm_p2_2; dy=norm_p2_2_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',colors(2,:));
xlim([min(q0s) max(q0s)]); ylim([-.01 1.01]); xlabel('boundary offset q_0')
PAPER testing the normal approximation for 4 4d t distributions
n_class=4; N_samp=1e4; n_samp=1e2;
% t distribution parameters
params_t=struct;
params_t(1).mu=[0;-4;-2;-2]; params_t(1).v=diag([1 2 3 4]);
params_t(2).mu=[1;1;0;0]; params_t(2).v=eye(4)/4;
params_t(3).mu=[3;4;5;7]; params_t(3).v=eye(4);
params_t(4).mu=[5;5;7;4]; params_t(4).v=eye(4);
samples=struct; params_samp=struct;
for i=1:4
% generate t samples
sample=mvtrnd(params_t(i).v,3,N_samp)+params_t(i).mu';
samples(i).sample=sample;
% mean and covariance of samples
params_samp(i).mu=mean(sample)';
params_samp(i).v=cov(sample)';
end
len_fam=10;
prior_fam=10.^linspace(-6,12,len_fam)';
vscale_fam=10.^linspace(-1,3,len_fam)';
true_p22_mean=nan(len_fam,1);
true_pe_mean=nan(len_fam,1);
true_pe_sd=nan(len_fam,1);
norm_p22_mean=nan(len_fam,1);
norm_pe_mean=nan(len_fam,1);
norm_pe_sd=nan(len_fam,1);
parfor i=1:len_fam
i
% generate family of classification domains
% uncomment when varying prior, comment when varying variance
priors_shift=[1 prior_fam(i) 1 1]; priors_shift=priors_shift/sum(priors_shift);
domains_shift=cell(n_class,1);
for k=1:n_class
domains_shift{k}=@(n,mu,v) opt_class_multi(n,params_samp,k,'mu',mu,'v',v,'priors',priors_shift);
end
% uncomment when varying variance, comment when varying prior
% params_samp_shift=params_samp;
% for k=1:n_class
% params_samp_shift(k).v=vscale_fam(i)*params_samp(k).v;
% end
% domains_shift=cell(n_class,1);
% for k=1:n_class
% domains_shift{k}=@(n,mu,v) opt_class_multi(n,params_samp_shift,k,'mu',mu,'v',v);
% end
results_shift=classify_normals_multi(samples,'input_type','samp','doms',domains_shift,'mc_samples',5e3,'plotmode',false);
% true error rates
true_p22_mean(i)=results_shift.samp_errmat(2,2)/N_samp;
true_pe_mean(i)=results_shift.samp_err;
samp_errmat_p=results_shift.samp_errmat/(n_class*N_samp);
errs=sum(~eye(length(samples)).*samp_errmat_p,2);
accs=diag(samp_errmat_p);
true_pe_sd(i)=sqrt(sum(errs.*accs)/n_samp);
% normal error rates
norm_p22_mean(i)=results_shift.norm_errmat(2,2)/sum(results_shift.norm_errmat(2,:));
norm_pe_mean(i)=results_shift.norm_err;
errs=sum(~eye(length(samples)).*results_shift.norm_errmat,2);
accs=diag(results_shift.norm_errmat);
norm_pe_sd(i)=sqrt(sum(errs.*accs)/n_samp);
end
ans =
1
ans =
2
ans =
3
ans =
4
Integrating normal 1 in domain 1
Integrating normal 1 in domain 1
Integrating normal 1 in domain 1
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
ans =
6
Integrating normal 1 in domain 1
ans =
10
Integrating normal 1 in domain 1
ans =
5
Integrating normal 1 in domain 1
ans =
8
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
ans =
7
Integrating normal 1 in domain 1
ans =
9
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
true_p22_sd=sqrt(true_p22_mean.*(1-true_p22_mean)/n_samp);
norm_p22_sd=sqrt(norm_p22_mean.*(1-norm_p22_mean)/n_samp);
figure; hold on;
% uncomment when varying prior, comment when varying variance
xline(1/3,'k','optimal boundary','LabelVerticalAlignment','bottom')
xlabel('assumed prior ratio p_1/(p_2+p_3+p_4)')
x=prior_fam/3;
% uncomment when varying variance, comment when varying prior
% xline(1,'k','optimal boundary','LabelVerticalAlignment','bottom')
% xlabel('assumed variance scale')
% x=vscale_fam;
% p(2|2)
y=true_p22_mean; dy=true_p22_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],[0.9290 0.6940 0.1250],'facealpha',.3,'linestyle','none');
y=norm_p22_mean; dy=norm_p22_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',[0.9290 0.6940 0.1250]);
% pe
y=true_pe_mean; dy=true_pe_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],[0.6 0.4 0.2],'facealpha',.3,'linestyle','none');
y=norm_pe_mean; dy=norm_pe_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',[0.6 0.4 0.2]);
xlim([min(x) max(x)]); set(gca,'xscale','log'); ylim([-.01 1.01])
PAPER Actual vision research data: detecting targets on natural scenes
absent=importdata('target_absent.txt',',',1);
present=importdata('target_present.txt',',',1);
results=classify_normals(absent.data,present.data,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -4569
1 11 -4579 initial simplex
2 13 -4582 expand
3 14 -4582 reflect
4 15 -4582 reflect
5 16 -4582 reflect
6 17 -4582 reflect
7 19 -4583 reflect
8 20 -4583 reflect
9 21 -4583 reflect
10 23 -4585 reflect
11 24 -4585 reflect
12 25 -4585 reflect
13 27 -4586 reflect
14 28 -4586 reflect
15 29 -4586 reflect
16 41 -4586 shrink
17 42 -4586 reflect
18 44 -4586 contract inside
19 46 -4586 contract inside
20 48 -4586 contract inside
21 50 -4586 contract inside
22 52 -4586 contract inside
23 53 -4586 reflect
24 54 -4586 reflect
25 56 -4586 contract inside
26 68 -4587 shrink
27 70 -4587 contract inside
28 72 -4588 contract inside
29 73 -4588 reflect
30 74 -4588 reflect
31 86 -4588 shrink
32 98 -4588 shrink
33 99 -4588 reflect
34 100 -4588 reflect
35 101 -4588 reflect
36 102 -4588 reflect
37 114 -4588 shrink
38 116 -4588 contract outside
39 128 -4588 shrink
40 130 -4588 contract inside
41 142 -4588 shrink
42 154 -4588 shrink
43 166 -4588 shrink
44 178 -4588 shrink
45 190 -4588 shrink
46 202 -4588 shrink
47 214 -4588 shrink
48 226 -4588 shrink
49 238 -4588 shrink
50 250 -4588 shrink
51 262 -4588 shrink
52 274 -4588 shrink
53 286 -4588 shrink
54 298 -4588 shrink
55 310 -4588 shrink
56 322 -4588 shrink
57 334 -4588 shrink
58 346 -4588 shrink
59 358 -4588 shrink
60 370 -4588 shrink
61 382 -4588 shrink
62 394 -4588 shrink
63 406 -4588 shrink
64 418 -4588 shrink
65 430 -4588 shrink
66 442 -4588 shrink
67 454 -4588 shrink
68 466 -4588 shrink
69 478 -4588 shrink
70 490 -4588 shrink
71 502 -4588 shrink
72 514 -4588 shrink
73 526 -4588 shrink
74 538 -4588 shrink
75 550 -4588 shrink
76 562 -4588 shrink
77 574 -4588 shrink
78 586 -4588 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 2.173913e-04
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [3×3754 double]
norm_errmat: [2×2 double]
norm_err: 8.0400e-05
norm_dprime: 7.5475
norm_maha_dprime: 6.8175
samp_errmat: [2×2 double]
samp_err: 0.0067
samp_dprime: 4.9418
samp_dv: {[2300×1 double] [2300×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[2300×1 double] [2300×1 double]}
samp_opt_bd_pts: [3×26070 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.0026
samp_opt_dprime: 5.5866
axis normal
axis([-50 170 -320 200 0 1000]); view(166,40)
xlabel('template'); ylabel('silhouette'); zlabel('edge');
PAPER Actual vision research data: detecting camouflage
load camouflage_edge_data
results_joint_2=classify_normals([edge_powers_2(:,1),edge_lpr_2(:,1)],[edge_powers_2(:,2),edge_lpr_2(:,2)],'input_type','samp');
Iteration Func-count min f(x) Procedure
0 1 -1.99656e+06
1 7 -1.99716e+06 initial simplex
2 8 -1.99716e+06 reflect
3 10 -1.99716e+06 contract inside
4 12 -1.99716e+06 contract inside
5 14 -1.99716e+06 contract outside
6 16 -1.99716e+06 contract outside
7 18 -1.99716e+06 contract inside
8 20 -1.99716e+06 contract inside
9 21 -1.99716e+06 reflect
10 23 -1.99719e+06 contract inside
11 25 -1.99756e+06 contract inside
12 27 -1.99756e+06 contract inside
13 29 -1.99758e+06 contract inside
14 31 -1.99763e+06 contract inside
15 33 -1.99774e+06 contract inside
16 34 -1.99774e+06 reflect
17 36 -1.99774e+06 contract inside
18 38 -1.99778e+06 contract inside
19 39 -1.99778e+06 reflect
20 40 -1.99778e+06 reflect
21 42 -1.99778e+06 contract inside
22 43 -1.99778e+06 reflect
23 45 -1.99784e+06 contract inside
24 47 -1.99784e+06 contract inside
25 48 -1.99784e+06 reflect
26 50 -1.99784e+06 contract inside
27 52 -1.99784e+06 contract inside
28 54 -1.99784e+06 contract inside
29 55 -1.99784e+06 reflect
30 56 -1.99784e+06 reflect
31 58 -1.99784e+06 contract inside
32 60 -1.99784e+06 contract inside
33 62 -1.99785e+06 reflect
34 64 -1.99785e+06 contract outside
35 65 -1.99785e+06 reflect
36 67 -1.99785e+06 contract inside
37 69 -1.99785e+06 contract outside
38 70 -1.99785e+06 reflect
39 72 -1.99785e+06 contract inside
40 74 -1.99785e+06 contract inside
41 76 -1.99785e+06 contract inside
42 77 -1.99785e+06 reflect
43 79 -1.99785e+06 contract inside
44 81 -1.99785e+06 reflect
45 82 -1.99785e+06 reflect
46 83 -1.99785e+06 reflect
47 84 -1.99785e+06 reflect
48 86 -1.99785e+06 contract inside
49 88 -1.99785e+06 contract outside
50 90 -1.99785e+06 reflect
51 98 -1.99785e+06 shrink
52 100 -1.99785e+06 contract outside
53 102 -1.99785e+06 contract outside
54 104 -1.99785e+06 contract inside
55 106 -1.99786e+06 contract inside
56 107 -1.99786e+06 reflect
57 115 -1.99786e+06 shrink
58 123 -1.99786e+06 shrink
59 125 -1.99786e+06 contract inside
60 127 -1.99786e+06 contract outside
61 128 -1.99786e+06 reflect
62 129 -1.99786e+06 reflect
63 137 -1.99786e+06 shrink
64 138 -1.99786e+06 reflect
65 140 -1.99786e+06 contract inside
66 142 -1.99786e+06 contract inside
67 150 -1.99786e+06 shrink
68 151 -1.99786e+06 reflect
69 152 -1.99786e+06 reflect
70 153 -1.99786e+06 reflect
71 155 -1.99786e+06 contract inside
72 163 -1.99786e+06 shrink
73 171 -1.99786e+06 shrink
74 179 -1.99786e+06 shrink
75 187 -1.99786e+06 shrink
76 195 -1.99786e+06 shrink
77 203 -1.99786e+06 shrink
78 211 -1.99786e+06 shrink
79 219 -1.99786e+06 shrink
80 227 -1.99786e+06 shrink
81 235 -1.99786e+06 shrink
82 243 -1.99786e+06 shrink
83 251 -1.99786e+06 shrink
84 259 -1.99786e+06 shrink
85 267 -1.99786e+06 shrink
86 275 -1.99786e+06 shrink
87 283 -1.99786e+06 shrink
88 291 -1.99786e+06 shrink
89 299 -1.99786e+06 shrink
90 307 -1.99786e+06 shrink
91 315 -1.99786e+06 shrink
92 323 -1.99786e+06 shrink
93 331 -1.99786e+06 shrink
94 339 -1.99786e+06 shrink
95 347 -1.99786e+06 shrink
96 355 -1.99786e+06 shrink
97 363 -1.99786e+06 shrink
98 371 -1.99786e+06 shrink
99 379 -1.99786e+06 shrink
100 387 -1.99786e+06 shrink
101 395 -1.99786e+06 shrink
102 403 -1.99786e+06 shrink
103 411 -1.99786e+06 shrink
104 419 -1.99786e+06 shrink
105 427 -1.99786e+06 shrink
106 435 -1.99786e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
axis([0.2 1 -900 400])
xlabel 'edge power'; ylabel 'edge spectrum'
results_dv_2=classify_normals(results_joint_2.samp_opt_dv{1},results_joint_2.samp_opt_dv{2},'input_type','samp','dom',@(x) x,'dom_type','fun','samp_opt',false);
axis([-50 100 0 .04])
xlabel('$q_s(${\boldmath$x$}$)$','interpreter','latex');
results_joint_4=classify_normals([edge_powers_4(:,1),edge_lpr_4(:,1)],[edge_powers_4(:,2),edge_lpr_4(:,2)],'input_type','samp','plotmode',false);
Iteration Func-count min f(x) Procedure
0 1 -1.9574e+06
1 7 -1.9574e+06 initial simplex
2 9 -1.9574e+06 contract inside
3 11 -1.9574e+06 contract outside
4 13 -1.9574e+06 contract outside
5 14 -1.9574e+06 reflect
6 16 -1.9574e+06 contract inside
7 18 -1.9574e+06 contract inside
8 20 -1.9574e+06 contract outside
9 22 -1.9574e+06 contract inside
10 24 -1.9574e+06 contract inside
11 26 -1.9574e+06 contract inside
12 28 -1.95741e+06 contract inside
13 30 -1.95782e+06 contract inside
14 32 -1.95782e+06 contract outside
15 34 -1.95783e+06 contract inside
16 36 -1.95808e+06 contract inside
17 37 -1.95808e+06 reflect
18 39 -1.95808e+06 contract inside
19 41 -1.95808e+06 contract inside
20 43 -1.95808e+06 contract inside
21 45 -1.95808e+06 contract inside
22 47 -1.95808e+06 contract inside
23 49 -1.95811e+06 contract inside
24 50 -1.95811e+06 reflect
25 52 -1.95811e+06 contract inside
26 54 -1.95811e+06 contract inside
27 56 -1.95814e+06 contract inside
28 58 -1.95818e+06 contract inside
29 60 -1.95819e+06 reflect
30 62 -1.95819e+06 contract inside
31 64 -1.95819e+06 contract inside
32 66 -1.95819e+06 contract inside
33 68 -1.95819e+06 contract inside
34 70 -1.9582e+06 contract inside
35 72 -1.9582e+06 contract inside
36 74 -1.9582e+06 contract inside
37 75 -1.9582e+06 reflect
38 77 -1.95822e+06 contract inside
39 78 -1.95822e+06 reflect
40 80 -1.95822e+06 contract inside
41 82 -1.95822e+06 contract inside
42 84 -1.95822e+06 contract inside
43 86 -1.95822e+06 contract inside
44 88 -1.95822e+06 contract inside
45 89 -1.95822e+06 reflect
46 90 -1.95822e+06 reflect
47 92 -1.95823e+06 expand
48 93 -1.95823e+06 reflect
49 95 -1.95823e+06 contract inside
50 96 -1.95823e+06 reflect
51 97 -1.95823e+06 reflect
52 99 -1.95824e+06 reflect
53 101 -1.95824e+06 contract inside
54 103 -1.95825e+06 expand
55 104 -1.95825e+06 reflect
56 106 -1.95825e+06 reflect
57 107 -1.95825e+06 reflect
58 115 -1.95825e+06 shrink
59 117 -1.95825e+06 contract inside
60 118 -1.95825e+06 reflect
61 120 -1.95825e+06 contract inside
62 122 -1.95826e+06 expand
63 123 -1.95826e+06 reflect
64 125 -1.95826e+06 contract inside
65 127 -1.95826e+06 contract inside
66 128 -1.95826e+06 reflect
67 130 -1.95827e+06 expand
68 132 -1.95827e+06 contract outside
69 133 -1.95827e+06 reflect
70 134 -1.95827e+06 reflect
71 135 -1.95827e+06 reflect
72 137 -1.95828e+06 expand
73 138 -1.95828e+06 reflect
74 140 -1.95828e+06 reflect
75 142 -1.95829e+06 expand
76 143 -1.95829e+06 reflect
77 145 -1.95829e+06 contract inside
78 147 -1.95834e+06 expand
79 148 -1.95834e+06 reflect
80 149 -1.95834e+06 reflect
81 150 -1.95834e+06 reflect
82 151 -1.95834e+06 reflect
83 153 -1.95838e+06 expand
84 154 -1.95838e+06 reflect
85 155 -1.95838e+06 reflect
86 156 -1.95838e+06 reflect
87 158 -1.9584e+06 expand
88 159 -1.9584e+06 reflect
89 161 -1.95846e+06 expand
90 162 -1.95846e+06 reflect
91 163 -1.95846e+06 reflect
92 164 -1.95846e+06 reflect
93 166 -1.9585e+06 expand
94 167 -1.9585e+06 reflect
95 169 -1.95856e+06 expand
96 170 -1.95856e+06 reflect
97 172 -1.95862e+06 expand
98 173 -1.95862e+06 reflect
99 174 -1.95862e+06 reflect
100 175 -1.95862e+06 reflect
101 177 -1.95865e+06 expand
102 178 -1.95865e+06 reflect
103 179 -1.95865e+06 reflect
104 180 -1.95865e+06 reflect
105 182 -1.95867e+06 expand
106 184 -1.95867e+06 contract outside
107 186 -1.95872e+06 expand
108 188 -1.95872e+06 contract inside
109 189 -1.95872e+06 reflect
110 191 -1.95872e+06 contract inside
111 192 -1.95872e+06 reflect
112 194 -1.95878e+06 expand
113 195 -1.95878e+06 reflect
114 196 -1.95878e+06 reflect
115 197 -1.95878e+06 reflect
116 199 -1.95888e+06 expand
117 200 -1.95888e+06 reflect
118 201 -1.95888e+06 reflect
119 203 -1.95898e+06 expand
120 204 -1.95898e+06 reflect
121 205 -1.95898e+06 reflect
122 207 -1.9591e+06 expand
123 208 -1.9591e+06 reflect
124 209 -1.9591e+06 reflect
125 211 -1.95917e+06 expand
126 213 -1.95933e+06 expand
127 214 -1.95933e+06 reflect
128 215 -1.95933e+06 reflect
129 217 -1.95945e+06 expand
130 218 -1.95945e+06 reflect
131 219 -1.95945e+06 reflect
132 221 -1.95961e+06 expand
133 222 -1.95961e+06 reflect
134 223 -1.95961e+06 reflect
135 225 -1.95961e+06 contract inside
136 227 -1.95964e+06 reflect
137 229 -1.95969e+06 reflect
138 230 -1.95969e+06 reflect
139 232 -1.95969e+06 contract inside
140 234 -1.95969e+06 contract inside
141 235 -1.95969e+06 reflect
142 236 -1.95969e+06 reflect
143 237 -1.95969e+06 reflect
144 238 -1.95969e+06 reflect
145 240 -1.95969e+06 contract outside
146 242 -1.95969e+06 contract inside
147 244 -1.95971e+06 reflect
148 246 -1.95976e+06 expand
149 248 -1.95976e+06 contract outside
150 249 -1.95976e+06 reflect
151 250 -1.95976e+06 reflect
152 252 -1.95981e+06 expand
153 253 -1.95981e+06 reflect
154 254 -1.95981e+06 reflect
155 256 -1.95996e+06 expand
156 257 -1.95996e+06 reflect
157 259 -1.96005e+06 expand
158 260 -1.96005e+06 reflect
159 262 -1.96018e+06 expand
160 263 -1.96018e+06 reflect
161 265 -1.96059e+06 expand
162 266 -1.96059e+06 reflect
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165 270 -1.96078e+06 expand
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167 274 -1.96135e+06 expand
168 275 -1.96135e+06 reflect
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171 279 -1.96135e+06 contract inside
172 280 -1.96135e+06 reflect
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175 285 -1.96172e+06 expand
176 287 -1.96179e+06 reflect
177 288 -1.96179e+06 reflect
178 290 -1.96179e+06 contract inside
179 291 -1.96179e+06 reflect
180 293 -1.96179e+06 contract inside
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182 296 -1.96179e+06 reflect
183 298 -1.96179e+06 contract inside
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186 304 -1.96188e+06 reflect
187 306 -1.96188e+06 contract inside
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190 311 -1.9619e+06 reflect
191 313 -1.9619e+06 contract outside
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195 319 -1.96195e+06 reflect
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198 324 -1.96195e+06 reflect
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203 331 -1.96195e+06 reflect
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206 336 -1.96195e+06 reflect
207 338 -1.96195e+06 contract inside
208 339 -1.96195e+06 reflect
209 341 -1.96195e+06 contract inside
210 342 -1.96195e+06 reflect
211 350 -1.96196e+06 shrink
212 351 -1.96196e+06 reflect
213 353 -1.96196e+06 contract outside
214 354 -1.96196e+06 reflect
215 356 -1.96196e+06 contract inside
216 358 -1.96196e+06 contract outside
217 359 -1.96196e+06 reflect
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219 362 -1.96196e+06 contract inside
220 364 -1.96196e+06 contract inside
221 365 -1.96196e+06 reflect
222 367 -1.96196e+06 contract inside
223 368 -1.96196e+06 reflect
224 376 -1.96196e+06 shrink
225 377 -1.96196e+06 reflect
226 379 -1.96196e+06 contract outside
227 381 -1.96196e+06 contract inside
228 389 -1.96197e+06 shrink
229 391 -1.96197e+06 contract inside
230 392 -1.96197e+06 reflect
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232 395 -1.96197e+06 contract outside
233 396 -1.96197e+06 reflect
234 398 -1.96197e+06 contract inside
235 399 -1.96197e+06 reflect
236 400 -1.96197e+06 reflect
237 401 -1.96197e+06 reflect
238 409 -1.96197e+06 shrink
239 411 -1.96197e+06 contract inside
240 412 -1.96197e+06 reflect
241 414 -1.96197e+06 contract inside
242 415 -1.96197e+06 reflect
243 423 -1.96197e+06 shrink
244 424 -1.96197e+06 reflect
245 426 -1.96197e+06 contract outside
246 434 -1.96197e+06 shrink
247 442 -1.96197e+06 shrink
248 444 -1.96197e+06 contract inside
249 446 -1.96197e+06 contract inside
250 454 -1.96197e+06 shrink
251 462 -1.96197e+06 shrink
252 470 -1.96197e+06 shrink
253 478 -1.96197e+06 shrink
254 486 -1.96197e+06 shrink
255 494 -1.96197e+06 shrink
256 502 -1.96197e+06 shrink
257 510 -1.96197e+06 shrink
258 518 -1.96197e+06 shrink
259 526 -1.96197e+06 shrink
260 534 -1.96197e+06 shrink
261 542 -1.96197e+06 shrink
262 550 -1.96197e+06 shrink
263 558 -1.96197e+06 shrink
264 566 -1.96197e+06 shrink
265 574 -1.96197e+06 shrink
266 582 -1.96197e+06 shrink
267 590 -1.96197e+06 shrink
268 598 -1.96197e+06 shrink
269 606 -1.96197e+06 shrink
270 614 -1.96197e+06 shrink
271 622 -1.96197e+06 shrink
272 630 -1.96197e+06 shrink
273 638 -1.96197e+06 shrink
274 646 -1.96197e+06 shrink
275 654 -1.96197e+06 shrink
276 662 -1.96197e+06 shrink
277 670 -1.96197e+06 shrink
278 678 -1.96197e+06 shrink
279 686 -1.96197e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
results_joint_8=classify_normals([edge_powers_8(:,1),edge_lpr_8(:,1)],[edge_powers_8(:,2),edge_lpr_8(:,2)],'input_type','samp','plotmode',false);
Iteration Func-count min f(x) Procedure
0 1 -1.78592e+06
1 7 -1.78592e+06 initial simplex
2 9 -1.78592e+06 contract inside
3 11 -1.78592e+06 contract inside
4 13 -1.78592e+06 contract inside
5 15 -1.78699e+06 contract outside
6 17 -1.78713e+06 contract inside
7 19 -1.78734e+06 contract outside
8 21 -1.78746e+06 contract inside
9 23 -1.78777e+06 contract inside
10 25 -1.78777e+06 contract inside
11 27 -1.7878e+06 contract inside
12 29 -1.78788e+06 contract outside
13 31 -1.78788e+06 contract outside
14 33 -1.78792e+06 contract inside
15 35 -1.78837e+06 reflect
16 36 -1.78837e+06 reflect
17 38 -1.78837e+06 contract inside
18 39 -1.78837e+06 reflect
19 41 -1.78837e+06 contract inside
20 43 -1.78837e+06 contract inside
21 44 -1.78837e+06 reflect
22 46 -1.78893e+06 expand
23 47 -1.78893e+06 reflect
24 48 -1.78893e+06 reflect
25 49 -1.78893e+06 reflect
26 50 -1.78893e+06 reflect
27 51 -1.78893e+06 reflect
28 53 -1.78903e+06 expand
29 54 -1.78903e+06 reflect
30 56 -1.78903e+06 contract inside
31 58 -1.78924e+06 expand
32 59 -1.78924e+06 reflect
33 61 -1.7893e+06 reflect
34 63 -1.78992e+06 expand
35 64 -1.78992e+06 reflect
36 66 -1.78992e+06 contract inside
37 67 -1.78992e+06 reflect
38 69 -1.78996e+06 expand
39 71 -1.78999e+06 reflect
40 73 -1.79056e+06 expand
41 74 -1.79056e+06 reflect
42 76 -1.79157e+06 expand
43 77 -1.79157e+06 reflect
44 78 -1.79157e+06 reflect
45 79 -1.79157e+06 reflect
46 81 -1.79183e+06 reflect
47 83 -1.79314e+06 expand
48 84 -1.79314e+06 reflect
49 85 -1.79314e+06 reflect
50 87 -1.79341e+06 expand
51 89 -1.79449e+06 expand
52 91 -1.79567e+06 expand
53 92 -1.79567e+06 reflect
54 93 -1.79567e+06 reflect
55 95 -1.79608e+06 expand
56 97 -1.79781e+06 expand
57 99 -1.79889e+06 expand
58 101 -1.79889e+06 contract inside
59 102 -1.79889e+06 reflect
60 103 -1.79889e+06 reflect
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62 105 -1.79889e+06 reflect
63 107 -1.79889e+06 contract inside
64 108 -1.79889e+06 reflect
65 110 -1.79907e+06 reflect
66 112 -1.79907e+06 contract outside
67 114 -1.79907e+06 contract inside
68 115 -1.79907e+06 reflect
69 116 -1.79907e+06 reflect
70 118 -1.79914e+06 contract inside
71 120 -1.79914e+06 contract inside
72 122 -1.79927e+06 reflect
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74 125 -1.79927e+06 contract inside
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76 128 -1.79927e+06 reflect
77 130 -1.79927e+06 contract inside
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81 137 -1.79928e+06 reflect
82 139 -1.79928e+06 contract inside
83 140 -1.79928e+06 reflect
84 142 -1.79928e+06 contract outside
85 144 -1.79928e+06 contract inside
86 145 -1.79928e+06 reflect
87 146 -1.79928e+06 reflect
88 148 -1.79928e+06 contract inside
89 149 -1.79928e+06 reflect
90 150 -1.79928e+06 reflect
91 158 -1.79928e+06 shrink
92 160 -1.79928e+06 contract outside
93 161 -1.79928e+06 reflect
94 162 -1.79928e+06 reflect
95 164 -1.79931e+06 expand
96 166 -1.79931e+06 contract inside
97 168 -1.79931e+06 contract inside
98 176 -1.79931e+06 shrink
99 177 -1.79931e+06 reflect
100 178 -1.79931e+06 reflect
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103 182 -1.79931e+06 contract inside
104 190 -1.79932e+06 shrink
105 191 -1.79932e+06 reflect
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107 194 -1.79932e+06 contract inside
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110 200 -1.79932e+06 contract inside
111 208 -1.79932e+06 shrink
112 210 -1.79932e+06 contract outside
113 212 -1.79932e+06 contract inside
114 213 -1.79932e+06 reflect
115 215 -1.79933e+06 reflect
116 217 -1.79933e+06 contract inside
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118 227 -1.79933e+06 shrink
119 229 -1.79933e+06 contract inside
120 231 -1.79933e+06 contract inside
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122 234 -1.79933e+06 reflect
123 236 -1.79933e+06 contract inside
124 237 -1.79933e+06 reflect
125 239 -1.79933e+06 contract inside
126 240 -1.79933e+06 reflect
127 248 -1.79933e+06 shrink
128 249 -1.79933e+06 reflect
129 250 -1.79933e+06 reflect
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132 254 -1.79933e+06 contract inside
133 255 -1.79933e+06 reflect
134 257 -1.79933e+06 contract outside
135 265 -1.79933e+06 shrink
136 267 -1.79933e+06 contract outside
137 268 -1.79933e+06 reflect
138 270 -1.79933e+06 contract inside
139 272 -1.79933e+06 contract inside
140 274 -1.79933e+06 contract inside
141 282 -1.79933e+06 shrink
142 284 -1.79933e+06 contract inside
143 286 -1.79933e+06 contract inside
144 288 -1.79933e+06 contract inside
145 296 -1.79933e+06 shrink
146 298 -1.79933e+06 contract inside
147 306 -1.79933e+06 shrink
148 314 -1.79933e+06 shrink
149 322 -1.79933e+06 shrink
150 330 -1.79933e+06 shrink
151 338 -1.79933e+06 shrink
152 346 -1.79933e+06 shrink
153 354 -1.79933e+06 shrink
154 362 -1.79933e+06 shrink
155 363 -1.79933e+06 reflect
156 365 -1.79933e+06 contract outside
157 373 -1.79933e+06 shrink
158 381 -1.79933e+06 shrink
159 389 -1.79933e+06 shrink
160 397 -1.79933e+06 shrink
161 405 -1.79933e+06 shrink
162 413 -1.79933e+06 shrink
163 421 -1.79933e+06 shrink
164 429 -1.79933e+06 shrink
165 437 -1.79933e+06 shrink
166 445 -1.79933e+06 shrink
167 453 -1.79933e+06 shrink
168 461 -1.79933e+06 shrink
169 469 -1.79933e+06 shrink
170 477 -1.79933e+06 shrink
171 485 -1.79933e+06 shrink
172 493 -1.79933e+06 shrink
173 501 -1.79933e+06 shrink
174 509 -1.79933e+06 shrink
175 517 -1.79933e+06 shrink
176 525 -1.79933e+06 shrink
177 533 -1.79933e+06 shrink
178 541 -1.79933e+06 shrink
179 549 -1.79933e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
results_dv_joint=classify_normals([results_joint_2.samp_opt_dv{1},results_joint_4.samp_opt_dv{1},results_joint_8.samp_opt_dv{1}],[results_joint_2.samp_opt_dv{2},results_joint_4.samp_opt_dv{2},results_joint_8.samp_opt_dv{2}],'input_type','samp');
Iteration Func-count min f(x) Procedure
0 1 -1.99145e+06
1 11 -1.9929e+06 initial simplex
2 12 -1.9929e+06 reflect
3 13 -1.9929e+06 reflect
4 14 -1.9929e+06 reflect
5 15 -1.9929e+06 reflect
6 16 -1.9929e+06 reflect
7 17 -1.9929e+06 reflect
8 19 -1.99367e+06 expand
9 20 -1.99367e+06 reflect
10 22 -1.99445e+06 expand
11 23 -1.99445e+06 reflect
12 24 -1.99445e+06 reflect
13 25 -1.99445e+06 reflect
14 26 -1.99445e+06 reflect
15 27 -1.99445e+06 reflect
16 29 -1.99523e+06 expand
17 30 -1.99523e+06 reflect
18 32 -1.99604e+06 expand
19 33 -1.99604e+06 reflect
20 34 -1.99604e+06 reflect
21 35 -1.99604e+06 reflect
22 36 -1.99604e+06 reflect
23 37 -1.99604e+06 reflect
24 39 -1.99682e+06 expand
25 40 -1.99682e+06 reflect
26 41 -1.99682e+06 reflect
27 42 -1.99682e+06 reflect
28 43 -1.99682e+06 reflect
29 45 -1.99721e+06 expand
30 46 -1.99721e+06 reflect
31 47 -1.99721e+06 reflect
32 48 -1.99721e+06 reflect
33 49 -1.99721e+06 reflect
34 50 -1.99721e+06 reflect
35 51 -1.99721e+06 reflect
36 53 -1.99734e+06 expand
37 54 -1.99734e+06 reflect
38 55 -1.99734e+06 reflect
39 57 -1.99782e+06 expand
40 58 -1.99782e+06 reflect
41 59 -1.99782e+06 reflect
42 60 -1.99782e+06 reflect
43 61 -1.99782e+06 reflect
44 63 -1.99782e+06 contract outside
45 64 -1.99782e+06 reflect
46 65 -1.99782e+06 reflect
47 67 -1.99796e+06 expand
48 69 -1.99796e+06 contract inside
49 70 -1.99796e+06 reflect
50 71 -1.99796e+06 reflect
51 72 -1.99796e+06 reflect
52 74 -1.99806e+06 expand
53 75 -1.99806e+06 reflect
54 76 -1.99806e+06 reflect
55 78 -1.99806e+06 reflect
56 80 -1.99809e+06 expand
57 81 -1.99809e+06 reflect
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59 84 -1.9981e+06 reflect
60 85 -1.9981e+06 reflect
61 86 -1.9981e+06 reflect
62 87 -1.9981e+06 reflect
63 89 -1.9981e+06 contract inside
64 91 -1.9981e+06 contract inside
65 93 -1.9981e+06 contract inside
66 94 -1.9981e+06 reflect
67 96 -1.9981e+06 contract inside
68 98 -1.9981e+06 contract inside
69 99 -1.9981e+06 reflect
70 101 -1.9981e+06 contract inside
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72 104 -1.9981e+06 reflect
73 106 -1.9981e+06 contract inside
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75 110 -1.9981e+06 reflect
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77 113 -1.9981e+06 contract inside
78 115 -1.9981e+06 contract inside
79 116 -1.9981e+06 reflect
80 118 -1.99811e+06 reflect
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82 120 -1.99811e+06 reflect
83 121 -1.99811e+06 reflect
84 123 -1.99811e+06 contract inside
85 125 -1.99811e+06 contract outside
86 126 -1.99811e+06 reflect
87 127 -1.99811e+06 reflect
88 128 -1.99811e+06 reflect
89 130 -1.99811e+06 contract inside
90 132 -1.99812e+06 expand
91 134 -1.99812e+06 contract outside
92 136 -1.99812e+06 contract inside
93 138 -1.99812e+06 contract inside
94 139 -1.99812e+06 reflect
95 140 -1.99812e+06 reflect
96 142 -1.99812e+06 contract inside
97 144 -1.99812e+06 contract inside
98 145 -1.99812e+06 reflect
99 146 -1.99812e+06 reflect
100 158 -1.99812e+06 shrink
101 160 -1.99812e+06 contract outside
102 161 -1.99812e+06 reflect
103 162 -1.99812e+06 reflect
104 164 -1.99812e+06 contract outside
105 166 -1.99812e+06 contract inside
106 168 -1.99812e+06 contract inside
107 170 -1.99812e+06 contract outside
108 171 -1.99812e+06 reflect
109 172 -1.99812e+06 reflect
110 174 -1.99812e+06 contract inside
111 176 -1.99812e+06 contract inside
112 178 -1.99812e+06 contract outside
113 180 -1.99812e+06 contract inside
114 181 -1.99812e+06 reflect
115 193 -1.99812e+06 shrink
116 194 -1.99812e+06 reflect
117 195 -1.99812e+06 reflect
118 196 -1.99812e+06 reflect
119 208 -1.99812e+06 shrink
120 210 -1.99812e+06 contract outside
121 212 -1.99812e+06 contract outside
122 213 -1.99812e+06 reflect
123 215 -1.99812e+06 contract outside
124 227 -1.99812e+06 shrink
125 228 -1.99812e+06 reflect
126 229 -1.99812e+06 reflect
127 230 -1.99812e+06 reflect
128 231 -1.99812e+06 reflect
129 232 -1.99812e+06 reflect
130 233 -1.99812e+06 reflect
131 235 -1.99812e+06 contract outside
132 247 -1.99812e+06 shrink
133 248 -1.99812e+06 reflect
134 260 -1.99812e+06 shrink
135 272 -1.99812e+06 shrink
136 273 -1.99812e+06 reflect
137 285 -1.99812e+06 shrink
138 287 -1.99812e+06 contract inside
139 299 -1.99812e+06 shrink
140 311 -1.99812e+06 shrink
141 323 -1.99812e+06 shrink
142 335 -1.99812e+06 shrink
143 347 -1.99812e+06 shrink
144 359 -1.99812e+06 shrink
145 371 -1.99812e+06 shrink
146 383 -1.99812e+06 shrink
147 395 -1.99812e+06 shrink
148 407 -1.99812e+06 shrink
149 419 -1.99812e+06 shrink
150 431 -1.99812e+06 shrink
151 443 -1.99812e+06 shrink
152 455 -1.99812e+06 shrink
153 467 -1.99812e+06 shrink
154 479 -1.99812e+06 shrink
155 491 -1.99812e+06 shrink
156 503 -1.99812e+06 shrink
157 515 -1.99812e+06 shrink
158 527 -1.99812e+06 shrink
159 539 -1.99812e+06 shrink
160 551 -1.99812e+06 shrink
161 563 -1.99812e+06 shrink
162 575 -1.99812e+06 shrink
163 587 -1.99812e+06 shrink
164 599 -1.99812e+06 shrink
165 611 -1.99812e+06 shrink
166 623 -1.99812e+06 shrink
167 635 -1.99812e+06 shrink
168 647 -1.99812e+06 shrink
169 659 -1.99812e+06 shrink
170 671 -1.99812e+06 shrink
171 683 -1.99812e+06 shrink
172 695 -1.99812e+06 shrink
173 707 -1.99812e+06 shrink
174 719 -1.99812e+06 shrink
175 731 -1.99812e+06 shrink
176 743 -1.99812e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
axis image; axis([-100 125 -50 75 -50 50]); view(-20,24)
xlabel('$q_s(${\boldmath$x$}$)$ (2px)','interpreter','latex');
ylabel('$q_s(${\boldmath$x$}$)$ (4px)','interpreter','latex');
zlabel('$q_s(${\boldmath$x$}$)$ (8px)','interpreter','latex');
results_all=classify_normals([edge_powers_2(:,1),edge_lpr_2(:,1),...
edge_powers_4(:,1),edge_lpr_4(:,1),...
edge_powers_8(:,1),edge_lpr_8(:,1)],...
[edge_powers_2(:,2),edge_lpr_2(:,2),...
edge_powers_4(:,2),edge_lpr_4(:,2),...
edge_powers_8(:,2),edge_lpr_8(:,2)],'input_type','samp');
Iteration Func-count min f(x) Procedure
0 1 -1.99471e+06
1 29 -1.99689e+06 initial simplex
2 30 -1.99689e+06 reflect
3 31 -1.99689e+06 reflect
4 32 -1.99689e+06 reflect
5 34 -1.99689e+06 contract inside
6 36 -1.99689e+06 contract inside
7 38 -1.99689e+06 contract inside
8 40 -1.99689e+06 contract inside
9 41 -1.99689e+06 reflect
10 43 -1.99689e+06 contract inside
11 45 -1.99689e+06 contract inside
12 46 -1.99689e+06 reflect
13 48 -1.99689e+06 contract inside
14 50 -1.99689e+06 contract inside
15 52 -1.99689e+06 contract inside
16 54 -1.99691e+06 contract inside
17 56 -1.99718e+06 reflect
18 57 -1.99718e+06 reflect
19 59 -1.99721e+06 contract inside
20 61 -1.99749e+06 reflect
21 63 -1.99749e+06 contract inside
22 64 -1.99749e+06 reflect
23 65 -1.99749e+06 reflect
24 66 -1.99749e+06 reflect
25 67 -1.99749e+06 reflect
26 68 -1.99749e+06 reflect
27 69 -1.99749e+06 reflect
28 70 -1.99749e+06 reflect
29 72 -1.99749e+06 contract inside
30 74 -1.99749e+06 contract inside
31 76 -1.99749e+06 contract inside
32 78 -1.99749e+06 contract inside
33 80 -1.99749e+06 contract inside
34 82 -1.99749e+06 contract inside
35 84 -1.99749e+06 contract inside
36 86 -1.99749e+06 contract inside
37 88 -1.99749e+06 contract inside
38 90 -1.99749e+06 contract inside
39 91 -1.99749e+06 reflect
40 93 -1.99749e+06 contract outside
41 94 -1.99749e+06 reflect
42 96 -1.99749e+06 contract inside
43 98 -1.99749e+06 contract inside
44 100 -1.99749e+06 contract inside
45 101 -1.99749e+06 reflect
46 102 -1.99749e+06 reflect
47 103 -1.99749e+06 reflect
48 104 -1.99749e+06 reflect
49 106 -1.99749e+06 contract inside
50 108 -1.99749e+06 contract inside
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54 116 -1.99749e+06 contract inside
55 118 -1.99749e+06 contract inside
56 119 -1.99749e+06 reflect
57 121 -1.99749e+06 contract inside
58 122 -1.99749e+06 reflect
59 124 -1.99749e+06 contract outside
60 126 -1.99749e+06 contract inside
61 128 -1.99752e+06 contract inside
62 130 -1.99752e+06 contract inside
63 132 -1.99752e+06 contract inside
64 133 -1.99752e+06 reflect
65 135 -1.99752e+06 contract outside
66 137 -1.99752e+06 contract inside
67 139 -1.99752e+06 contract inside
68 141 -1.99752e+06 contract inside
69 143 -1.99754e+06 contract inside
70 145 -1.99757e+06 contract inside
71 146 -1.99757e+06 reflect
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756 1177 -1.99833e+06 contract inside
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760 1212 -1.99833e+06 shrink
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794 2232 -1.99833e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
results_all_samp_opt=classify_normals([edge_powers_2(:,1),edge_lpr_2(:,1),...
edge_powers_4(:,1),edge_lpr_4(:,1),...
edge_powers_8(:,1),edge_lpr_8(:,1)],...
[edge_powers_2(:,2),edge_lpr_2(:,2),...
edge_powers_4(:,2),edge_lpr_4(:,2),...
edge_powers_8(:,2),edge_lpr_8(:,2)],'input_type','samp',...
'dom',results_all.samp_opt_bd,'samp_opt',false);
xlabel('$q_s(${\boldmath$x$}$)$ (2, 4 and 8px)','interpreter','latex');
axis([-125 150 0 .015])
